Prior Performance and Risk-Taking of Mutual Fund Managers:

A Dynamic Bayesian Network Approach

Manuel Ammann and Michael Verhofen∗

October 2006

forthcoming in the Journal of Behavioral Finance, 8(1), 2007

Abstract

We analyze the behavior of mutual fund managers with a special focus on the impact of

prior performance. In contrast to previous studies, we do not solely focus on the volatility

as a measure of risk, but also consider alternative definitions of risk and style. Using a

Dynamic Bayesian Network, we are able to capture non-linear effects and to assign exact

probabilities to the mutual fund managers' adjustment of behavior. In contrast to theoretical

predictions and some existing studies, we find that prior performance has a positive impact

on the choice of the risk level, i.e., successful fund managers take more risk in the

following calendar year. In particular, they increase the volatility, the beta and the tracking

error and assign a higher proportion of their portfolio to value stocks, small firms and

momentum stocks. Overall, poor performing fund managers switch to passive strategies.

Keywords: Mutual Funds, Risk Taking, Dynamic Bayesian Network JEL classification: G21

∗ Manuel Ammann (manuel.ammann@unisg.ch) is professor of finance at the Swiss Institute of Banking and Finance, University of St. Gallen, Switzerland, and Michael Verhofen (verhofen@gmail.com) is visiting scholar at the Haas School of Business, University of California at Berkeley. We thank the editor, Bernd Brommundt, Alexander Ising, Stephan Kessler, Axel Kind, Jennifer Noll, Angelika Noll, Ralf Seiz, Stephan Süss, Rico von Wyss, and Andreas Zingg for valuable comments. We acknowledge helpful comments of the participants from the Joint Research Workshop of the University of St. Gallen and the University of Ulm in 2005. We acknowledge financial support from the Swiss National Science Foundation (SNF).

1 Introduction

The behavior of mutual fund managers has been subject to considerable academic research.

As rational agents, they are supposed to adjust their behavior in consideration of the

incentives they are facing. Basically, these incentives can be divided into two different

aspects, the structure of compensation schemes and the behavior of investors. On the one

hand, most compensation schemes are constructed like a call option, i.e., fund managers

have a higher upside than downside potential. This provides a temptation for fund

managers to increase the riskiness in a suboptimal way for the investor. On the other hand,

in a multi-period context, a positive relation between past performance and new fund flows

has been observed. A fund manager who has shown a high performance is rewarded with

new capital. In contrast, a manager with a poor performance does not suffer from the same

amount of cash outflows. If compensation is linked to fund size, this provides an incentive

for increasing the riskiness of a portfolio to a suboptimal point from an investor’s

perspective. In short, in so called mutual fund tournaments portfolio managers compete for

better performance, greater fund inflows, and higher compensation.

From a theoretical point of view, a number of authors focus on agency conflicts in the

mutual fund industry, i.e., on asymmetric information and hidden action between mutual

fund managers and their investors, and on the call option-like function of compensation

schemes. Fund managers might unnecessarily shift the fund’s risk in response to the fund’s

relative performance. This behavior is linked to compensation and investor’s reaction.

Carpenter (2000) solves the dynamic investment problem of a risk averse manager

compensated using a call option on the assets he controls, i.e., a convex compensation

scheme. She shows that under the manager’s optimal policy, the option is likely to end up

deep in or deep out of the money because managers take, in general, more risk than the

2

investor would choose. Berk & Green (2004) propose a model that incorporates two

important features: First, performance is not persistent and, second, fund flows rationally

respond to past performance. In particular, they assume that investors behave as Bayesians,

i.e., they update their beliefs about a fund manager’s skill based on observed returns and

prior beliefs. They show that a rational model for active portfolio management when talent

is a scarce resource disappearing as a fund grows can explain many empirical observations

without relying on investors irrationality and asymmetric information. Similar models also

relying on Bayesian updating have been proposed by Schmidt (2003) and Dangl, Wu &

Zechner (2004). Lynch & Musto (2003) address the question how fund managers change

their strategy over time. In their model, they show that strategy changes only occur after

periods of poor performance.

From an empirical point of view, different authors have analyzed the actual behavior of

mutual fund managers. Deli (2002) investigates marginal compensation rates in mutual

fund advisory contracts. He finds that marginal compensation depends positively on

turnover and the fund type (e.g. equity, closed-end), and is negatively related to fund size

and size of the fund family. Therefore, incentives to take risk might differ across fund

managers.

Chevalier & Ellison (1997) estimate the shape of the relationship between performance and

new fund flows because the shape of this relation creates incentives for fund managers to

increase or decrease the riskiness of the fund. They find that funds tend to change their

volatility depending on the relative performance by the end of September. Similarly,

Brown, Harlow & Starks (1996) focus on mid-year effects. In particular, they test the

hypothesis that mutual managers showing an underperformance by mid year change the

fund’s risk differently than those mutual fund managers showing an outperformance at the

3

same point in time. Their empirical analysis shows that mid-year losers tend to increase the

fund’s volatility to a greater extent than their successful counterparts. Busse (2001)

suggests that some prior findings might be spurious. Using daily data and the methodology

by Brown et al. (1996) he finds no mid-year effect. In sum, existing empirical analyses

have failed to deliver clear evidence on the behavior of mutual fund managers and there

are doubts about the robustness of the findings.

In this paper, we contribute to the existing empirical literature in a number of ways. In

contrast to existing studies, we do not solely focus on volatility as a measure of risk. We

take into account also other measures such as the beta and the tracking error, and style

measures such as the high-minus-low (HML) factor, the small-minus-big (SMB) factor,

and the momentum (UMD) factor. Furthermore, in contrast to previous studies, we use a

robust, non-parametric approach. As we do not impose any distributional assumptions, we

are able to capture a wide range of non-linear and asymmetric patterns. Moreover, instead

of using a particular theoretical model describing the behavior of fund managers as a

starting point, our study is designed in an explorative way. Concerning the data, we do not

use only a sub-group of mutual funds, but a complete set of all US equity funds for a long

time period of data.

In this paper, we compute conditional transition matrices and compare whether the

conditional transition matrices are different for successful and unsuccessful mutual funds.

For the empirical analysis, we use a Bayesian Network. A Bayesian Network is a model

for representing conditional dependencies between a set of random variables. Up to now,

research on Bayesian Networks (BN) is mainly concentrated in statistics and computer

science, particularly, in artificial intelligence (Korb & Nicholson (2004), Neapolitan

(2004)), in pattern recognition (Duda, Hart & Stork (2001)) and expert systems (Jensen

4

(2001), Cowell, Dawid, Lauritzen & Spiegelhalter (2003)). However, although the term

“Bayesian Network” has not yet widely spread, special cases of Bayesian Networks are

widely used in economics and finance. In particular, many state space models such as the

Kalman filter and Hidden Markov models are Bayesian Networks with a simple

dependency structure. Moreover, many classical econometric approaches such as discrete

choice models and regressions are so-called Naive Bayesian Networks because of their

mono-causal dependency structure (Jordan, Ghahramani & Saul (1997)).

Bayesian Networks work as follows. Suppose there are three variables such as the tracking

error in period T, the return in period T, and the tracking error in period T + 1. Suppose

next that all variables are conditionally dependent, i.e., the tracking error in period T

affects the return in period T and the tracking error in period T + 1. Moreover, also the

return in period T affects the tracking error in T + 1. In classical econometrics, this

problem is determined as multicollinearity and might lead to identification problems.

Bayesian networks can deal with such complex settings and can help to overcome the

identification problem. For a discussion of a wide range of different settings of Bayesian

Networks, we refer to Pearl (2000).

Besides being a modern tool for identifying the impact of and magnitude of different

causal sources, using Bayesian Networks has a number of advantages over standard

econometric methods. For example, they allow the computation of exact conditional

probabilities to assess the magnitude of a factor. For example, we can analyze whether the

probabilities change from 50:50 to 60:40 or to 90:10 for a given evidence. Moreover,

Bayesian Networks are able to capture non-linear and asymmetric patterns.

Using Bayesian Networks on a set of US equity funds over about 20 years of data, we find

that prior performance has a positive impact on the choice of the risk level, i.e., successful

5

fund managers take more risk in the following time period. In particular, they increase the

volatility, the beta, and the tracking error and assign a higher proportional of their portfolio

to value stocks, small firms and momentum stocks.

The article is structured as follows. In Section 2, we outline the econometric approach. In

Section 3, we present our empirical results. Section 4 concludes.

2 Model

2.1 Introduction to Bayesian Networks

We start with a description of Bayesian Networks. For short, a Bayesian Network is a

graphical model for representing conditional dependencies between a set of random

variables. This includes, in particular, learning about conditional distributions and updating

beliefs about probability distributions for a target node given some observation for at least

one variable.

Figure 1 shows a Dynamic Bayesian Network. The structure is similar to the Bayesian

Networks which have been proposed by Pearl (2000) for causal discovery. The circles

denote the nodes, i.e., the variables in the Bayesian Network, and the arcs the conditional

dependency between two nodes. The conditional dependency between two nodes can be

basically of any type. The most frequently conditional probability distributions (CPD) used

are Gaussian distributions and multinomial (or tabular) distributions. In the example in

Figure 1 a binomial distribution is used.

We are using a Dynamic Bayesian Network to analyze the revision of behavior in the

mutual fund industry. In particular, we analyze how past performance affects a set of

6

variables which describe the risk taking behavior of mutual fund managers. Suppose a fund

manager can, at least to some degree, choose the tracking error of his portfolio, i.e., how

close he mimics the relevant index or whether he chooses a more active investing style. By

assumption, there is a fifty-fifty chance that the portfolio manager chooses a high or low

tracking error. Suppose next that the choice of the magnitude of the tracking error affects

the subsequent return. In particular, a low tracking error leads to a chance of 50% of a high

return and to 50% of a low return. In contrast, it is assumed, that a high tracking error leads

to a chance of 90% of a low return and 10% of a high return.

The tracking error in period T + 1 is an effect of two variables, the tracking error in period

T and the return in period T. These conditional dependencies have a straightforward

interpretation. On the one hand, it might be that there is some degree of persistency in the

behavior, i.e., a fund manager who has had a high tracking error in period T tends to

maintain a high tracking error. On the other hand, there might be some degree of learning

or revision of behavior.

Figure 1 shows how the Dynamic Bayesian Network can be used to update beliefs about

probability distributions. Suppose the investor knows that a fund had a superior

performance in period T. The question of inference is how this knowledge affects the belief

about the distribution of the tracking error in period T + 1. As shown in Figure 1, the

straightforward application of Bayes theorem leads to a probability of 91.50% for a high

tracking error and of 8.5% for a low tracking error.

Bayesian Networks are frequently also denoted as probabilistic networks (Cowell et al.

(2003)), Bayesian Artificial Intelligence (Korb & Nicholson (2004)) and Bayesian Belief

Networks (Duda et al. (2001)). As noted by Borgelt & Kruse (2002), Bayesian Networks

7

rely on the achievements of a number of other concepts and in particular on classification

and regression trees, naive Bayes classifiers, artificial neural networks and graph theory. A

number of other econometric approaches such as the Kalman Filter, Hidden Markov

Models and state space models in general can be regarded as special cases of Bayesian

Networks and, in particular, of Dynamic Bayesian Networks (DBN), as shown by Jordan et

al. (1997). Even regressions and discrete choice models can be incorporated in a Bayesian

Network structure such as in so-called Naive Bayes Nets. Bayesian Networks (BN) are a

very powerful tool to deal with uncertainty, incomplete information and complex

probabilistic structures. In particular, Bayesian Networks enable the extraction of

probabilistic structures from data and decision making in these structures. Therefore,

Bayesian Networks are well-suited for financial applications.

In this section, we address the three main issues associated with Bayesian Networks. First,

the question of representation, i.e., what is a Bayesian Network. Second, the question of

learning (estimation) of parameters of a Bayesian Network. Third, the question of

inference, i.e., how Bayesian Networks can be used to answer probabilistic questions. For

an extended coverage of the topic and for a coverage of decision making within Bayesian

Networks, we refer to Jensen (2001), Cowell et al. (2003), Korb & Nicholson (2004) and

Neapolitan (2004).

2.2 Representation

As defined by Jensen (2001), a Bayesian Network consists of a set of variables with

directed arcs between these variables. These variables form a directed acyclic graph

(DAG) and, for each arc that connects two variables, a potential table, i.e., a conditional

distribution, is defined.

8

Suppose there are four random variables, A, B, C, and D. Applying the chain rule, the joint

probability can be written as a product of conditional probabilities

=),,,( DCBAP ⋅),,|( CBADP ⋅),|( BACP ⋅)|( ABP )(AP . Suppose next, A and D are

conditionally independent, i.e., ⋅= ),|(),|,( CBAPCBDAP ),|( CBDP and B and C are

conditionally independent. Using Bayes theorem and the previous factorization, it can be

shown that

. ),|()|()()(

),|()|()()(),(

),,,(),|,(

dAdDCBDPACPBPAPCBDPACPBPAP

CBPDCBAPCBDAP

⋅⋅⋅∫∫

⋅⋅⋅=

=

Basically, three elementary theorems of probability, i.e., Bayes theorem, the chain rule,

and conditional dependency, are the building blocks for Bayesian Networks.

2.3. Learning

In the previous subsection, we assumed that the parameters, i.e., the conditional probability

distributions, are known. However, in most cases it is necessary to learn about the

parameters of a Bayesian Network based on a data set. For parameter learning, maximum

likelihood (ML) can be used. As noted by Ghahramani (2001), the likelihood decouples

into local terms involving each node and its parents. This simplifies the maximum

likelihood estimation because it reduces to number of local maximization problems.

Suppose a data set consists of M cases for each of the n nodes. Let

denote the vector of observations for a single case for all nodes in the network. Therefore,

the training data set d is given

( )′= )()(

1)( ,..., h

nhh ddd

{ })()2()1( ,...,, Mdddd = . Neapolitan (2004) shows that the

likelihood function is given by

9

),|()|( )()(

11i

hi

hi

M

h

n

i

padPdL θθ ∏∏==

=

where contains the values of the parents of the node X)(hipa i in the hth case and θ the

parameter set.

In this paper, we use a multinomial distribution. In this case, the maximization problem

simplifies to a closed-form solution if the data are complete. Then, the likelihood function

is given by

, )(111

⎟⎟⎠

⎞⎜⎜⎝

⎛= ∏∏∏

===

ijkiji

sijk

r

k

q

j

n

i

FEdL

where qi denotes the parents of node Xi and ri the number of different classes of the

multinomial distribution. Fijk denotes the distribution of node i conditional on the parent

node j where the value of node xi is equal to k. The exponent sijk denotes the number of

cases in which xi is equal to k. The proof of these results can be found in Neapolitan

(2004). Suppose next the conditional distributions Fijk have a Dirichlet distribution, i.e., a

generalized Beta distribution, with (prior) parameters and

Then, the likelihood is given by

,1ija ,...,2ija ,iijra ijkkij aN ∑=

.ijkkij sM ∑=

)()(

)()(

)(111 ijk

ijkijkr

kijij

ijq

j

n

i asa

MNN

dLii

Γ

+Γ

+Γ

Γ= ∏∏∏

===

where denotes the gamma function. The use of Dirichlet distributions as conditional

distribution has a couple of advantages and is not restrictive at all. The Dirichlet

distribution is the so called natural conjugate prior for the multinomial distribution, i.e., an

application of Bayes theorem with the Dirichlet distribution as prior distribution and the

multinomial distribution as likelihood leads to a closed form solution for the posterior

distribution which has the same functional form as the prior, i.e., a Dirichlet distribution.

(.)Γ

10

Therefore, the Dirichlet distribution is very helpful for Bayesian sequential analysis and

Bayesian updating. Moreover, the Dirichlet distribution has upper and lower bounds and is

therefore very helpful for modelling probabilities that cannot become greater than 1 or

lower than 0. Concerning prior information, we can incorporate prior information about

conditional dependencies or assign almost uninformative priors by setting all aijk to 1.

2.4. Inference

Suppose, the structure of a Bayesian Network and all conditional probability distributions

(CPD) are known and a researcher has evidence about at least one node for a new case.

The goal of probabilistic inference (also belief updating, belief propagation or

marginalization) is to update the marginal probabilities in the network to incorporate this

new evidence (Ghahramani (2001)). Formally, the task of inference is to find the posterior

distribution , where X denotes the query node and E the set of evidence

nodes.

)|( eExXP ==

By using the local structure of a Bayesian Network, it can be shown that belief updating

can be divided into the predictive support for X from evidence nodes connected to X

through its parents, U1,…,Um, and the diagnostic support for X from evidence nodes

connected to X through its children, Y1,…,Ym (Korb & Nicholson (2004)). Pearl’s message

passing algorithm (Pearl (1982)) shows how to update the posterior distribution Bel(X).

The derivation involves the repeated application of Bayes Theorem and the use of the

conditional independencies encoded in the network structure. The basic idea is that, at each

iteration of the algorithm, Bel(X) is updated locally using three parameters, λ(X), π(X) and

the conditional probability table (CPT), where λ(X) and π(X) are computed using the

messages received from the parents π and from the children λ of node X (Korb &

11

Nicholson (2004)). In Bayes Theorem, π plays the role of the prior and λ the role of the

likelihood.

The algorithm requires three types of parameters to be maintained. First, the current

strength of the predictive support π contributed by each incoming link , i.e., XU i →

),|()( \ XUiiX iEUPU =π where is all evidence connected to UXUi

E \ i except via X. Second,

the current strength of the diagnostic support λ contributed by each outgoing link

)|()(: \ XEPXYX XYYj jj=→ λ where is all evidence connected to YXY j

E \ j through its

parents except via X. Third, the fixed conditional probability distribution (CPD)

, i.e., the conditional distribution of node X is only dependent on its

parents.

),...,|( ni UUXP

Pearl’s message-passing algorithm consists of two steps. In the first step, i.e., belief

updating, messages arriving from the parents or the children of an activated node X and

lead to changes in belief parameters. In the second step, i.e., bottom-up and top-down

propagation, the activated node computes new messages for the parents λ and for the

children λ to send it into the appropriate direction.

In the first step, the posterior distribution of each activated node X proportional to the

messages from the parents )( iX Uπ and to the messages from its children )(XjYλ is

determined as follows

)()()( iii xxxBel παλ=

where )(),...,|()( 1,...,1

iXi

niuu

i uuuxPxn

ππ ∏∑= and

12

⎪⎩

⎪⎨

⎧

∏

==

otherwise )(another for is evidence if 0

is evidence if 1)(

iYj

j

i

i

xx

xXx

jλ

λ

and where α is a normalizing constant rendering .1)( ==∑ ix xXBeli

In the second step, the bottom-up and top-down propagation step, node X sends new λ

messages to its parents

)(),...,|()()(:

kXik

niiiku

ix

iX uuuxPxuki

πλλ ∏∑∑≠≠

=

and the new π messages to its children

⎪⎩

⎪⎨

⎧

=otherwise )(/)(

lueanother vafor is evidence if 0entered is valueevidence if 1

)(

iYi

j

i

iY

xxBelx

xx

j

j

λαπ

This procedure is repeated until a node has received all messages.

Inference methods can be distinguished into two main categories, exact inference and

approximate inference algorithms. The latter have been developed because probabilistic

inference can be a computationally hard problem for complex networks with a large

number of nodes because required computational power increases exponentially with the

number of parent nodes. Well-known algorithms for exact inference include variable

elimination, Pearl’s message-passing algorithm, the noisy-or-gate algorithm, and the

junction tree algorithm. For approximate inference, standard algorithms include likelihood

weighting, logic sampling and Markov Chain Monte Carlo (MCMC) (for an overview, see

Korb & Nicholson (2004) or Neapolitan (2004)).

2.5. Data

13

For the analysis, we use a complete sample about all US open-end equity funds containing

1923 funds. The data set has been provided by Reuters Lipper. For each single fund,

information about the fund’s launch date, the fund’s sector (Equity, International, Large

Cap, Mid Cap, and Small Cap), its style (Income, Core, Growth, Value), its annual fee and

its total assets per April, 30th, 2004, are available. Price information is available on a

monthly basis for 19 years, i.e., from December 1984 to December 2003. For the analysis,

we exclude international mutual funds and funds with less than two years of data. For

benchmarking purposes, we use the excess return on S&P 500 index as the market

portfolio and the 3-month Treasury bill rate as the risk-free rate. The data are from

Datastream. As a second benchmark, the Carhart (1997) four-factor model is used. The

data for the market risk premium, the size premium, the value premium and for the

momentum premium, are from the Fama and French data library.

Table 1 shows the descriptive statistics for the data used in this study. The number of funds

increased from 191 in year 1985 to 1478 in the year 2003. The average return across all

funds fluctuated substantially during that time with a minimum of -26.35% in 2002 and a

maximum of 28.26% in 2003. Similarly, the performance of single funds shows a high

degree of dispersion.

2.6. Performance Measurement

To measure a fund’s performance, a number of different approaches have been suggested

(e.g. Kothari & Warner (2001), Wermers (2000), Daniel, Grinblatt, Titman & Wermers

(1997)). To estimate the exposure towards the Fama & French (1993) risk factors and the

Carhart (1997) momentum factor, we run the following regression for each fund i and for

each calendar year t:

14

tiCarhartUMDtiSMBti

HMLtiCarharttiCarharttiCarharttfti

rUMDrSMB

rHMLrMRPrr

,,,,

,,,,,,,

ε

α

+⋅+⋅

+⋅+⋅+=−

where ri,t denotes the return of a fund i, rf,t the risk-free rate and εCarhart,i,t the regression

residual. The coefficients to be estimated are denoted by MRPCarhart,i,t, HMLi,t, SMBi,t and

UMDi,t, and risk premia by rCarhart, rHML, rSMB and rUMD. An analogous approach is used for

the risk exposure with respect to the S&P 500, i.e.,

. ,,500500,,500,,500,, tiSPSPtiSPtiSPtfti rMRPrr εα +⋅+=−

In the following analysis, we refer to titiRaw rR ,,, = as the unadjusted return of a fund, to

tiSPtiSPR ,,500,,500 α= as the risk-adjusted return using the S&P 500 as benchmark, and to

tiCarharttiCarhartR ,,,, α= as the risk-adjusted return using the Carhart four-factor model as

benchmark.

The tracking error measures a fund's deviation from a passive index. We define the

tracking error TE as the volatility σ of the residuals of the regressions on the index, i.e.,

. )( and )( ,,500,,500,,,, tiSPtiSPtiCarharttiCarhart TETE εσεσ ==

2.7. Implementation

Figure 2 shows the corresponding Dynamic Bayesian Network. For each fund, we estimate

for each year eight different factors describing the behavior of a mutual fund. In particular,

we use the standard deviation of returns, the beta against the S&P 500 and the Fama and

French market portfolio, the loading on the value vs. growth factor, the loading on the size

factor, the loading on the momentum factor, and the tracking error against the S&P 500

and the Fama and French market portfolio.

15

A likelihood ratio test was performed and showed that all arcs in the Bayesian Network are

highly significant.

To get robust results that are independent of restrictive distributional assumptions and to

incorporate non-linear behavior, we use a multinomial distribution with four classes where

variables are grouped into quartiles for each year.

The implementation of the Bayesian Network has been carried out using the “Bayes Net

Toolbox for MatLab”. For testing, we created a large number of different artificial data sets

and re-extracted the underlying probability distributions. In all instances, the underlying

probability distributions were recovered accurately.

The Bayesian Network has been initialized by setting all aijk to 1. Therefore, the analysis

incorporates no material prior information. The Bayesian Network is primarily used as an

econometric tool.

3. Empirical Results

The analysis is structured as follows. We first focus on the marginal distributions within a

time period that is the relation between risk and return. Then, we focus on intertemporal

relations of different measures of risk and style. Finally, we turn to the marginal

distributions of risk and style conditional on past performance.

Due to the large amount of empirical results, we focus in the empirical part on risk-

adjusted returns using the S&P 500 as benchmark in a one-factor model. If not stated

otherwise, results for unadjusted returns and returns adjusted with the Carhart four-factor

model are very similar.

3.1. Risk and Return

16

Table 2 shows the relation between the set of variables describing the risk exposure of

mutual funds and the following performance measured on a risk-adjusted basis.

Significance has been tested against a null hypotheses of no probabilistic relation, i.e., the

null hypothesis for a multinomial distribution with four classes is that the probability for

each class is 25%.

Table 2 is interpreted as follows. Suppose a fund has in one particular year a volatility in

the lowest quartile among all other funds. Then, the fund has a probability of 11.8% (first

row, first column) to achieve a risk-adjusted return in the lowest quartile among all other

funds. Analogously, the chance is 30.8% (first row, fourth column) to obtain a risk-

adjusted return in the highest quartile among all other funds.

We find that the higher the volatility, the more likely a fund has an extreme return in

quartile 1 (Q1) and quartile 4 (Q4). The transition probability of ending in the first quartile

of returns is only 11.8% for funds with a volatility in Q1 and 22.8% for funds with a

volatility in Q4. Similarly, funds with a low standard deviation in Q1 have a chance of

30.8% of reaching a high return in Q4 whereas funds with a high standard deviation in Q4

have a chance of 41.4%. Consequently, funds with a low risk level have a higher chance of

reaching a return centered around the mean. For example, a fund with a volatility in Q1 has

a chance for a return in Q2 of 28.9% whereas a fund with a high volatility fund has a

chance of 16.6%.

For the exposure to the market risk, i.e., the beta, the finding is reversed. Funds with a low

beta in Q1 have a chance of 42.6% of achieving a risk-adjusted return in Q4. High-beta

funds have only a transition probability to a return in Q4 of 33.0%. The data show that the

higher a fund’s beta is, the lower is its relative risk-adjusted return.

17

The style factors in the Carhart (1997) four-factor model, i.e., the value premium, the size

premium and the momentum premium (UMD), show the expected results. As documented

by Fama & French (1993), trading strategies based on size factors and valuation ratios

have historically earned superior returns. Value oriented funds, i.e., funds with a high

loading on the value premium (HMLT in Q4) had a chance for a return in Q4 of 44.2%. In

contrast, growth funds (HMLT in Q1) had a chance for a return in Q4 of 32.6%. Similarly,

small-caps funds, i.e., funds with SMBT in Q4, had a chance for a return in Q4 of 44.4%,

while for large cap stocks this value was only 24.2%. Even more evident is this findings

for momentum funds. Funds with a high exposure to momentum stocks (Q4) had a 49.4%

chance for a return in the highest quartile. Vice versa, for funds avoiding momentum

stocks (Q1), this chance is reduced to 17.0%.

The tracking error indicates the magnitude of active portfolio management of a fund

manager. The transition matrix shows that fund managers with a high tracking error (Q4)

are with 45.8% more likely of having a return in the highest quartile than fund managers

with a low tracking error (Q1) having a chance of 25.%. The data indicate that active

portfolio management has had some value.

3.2. Persistence in Risk Levels

In this section, we analyze the persistence in the choice of risk levels. Table 3 shows the

relation between a measure of risk, e.g., the standard deviation, in period T and the

successive period T + 1. Of special interest are the diagonal elements of the transition

matrix because they represent the degree of persistence in the choice of risk levels.

For the standard deviation the probabilities of staying in the same quartile, i.e., the

diagonal elements of the transition matrix, are 50.4%, 37.8%, 41.2%, 70.1%, respectively.

18

This means that a fund with a low return volatility has a chance of 50.4% of staying in the

lowest quartile, a fund with a volatility in the second quartile of 37.8% of staying in the

same quartile, and so on. In contrast, for the exposure towards market risk, the data show a

lower degree of persistence in behavior. For the beta against the S&P 500 only 46.0%,

31.0%, 36.4%, 54.1%, respectively, stay in the same class. Therefore, we conclude that the

fund managers change their market risk to a larger extent than the portfolio’s volatility and,

in general, the degree of persistence is larger for funds with a very low or a very high

exposure to a risk factor.

Similar are the findings for the style factors in the Carhart (1997) four-factor model. While

for the HML factor, the transition probabilities for staying in the same class are 49.7%.

34.4%, 32.4%, 44.4%, the transition probabilities for the SMB factor are 49.03%, 33.1%,

35.1%, 68.7%. For the momentum (UMD) factor, the transition probabilities are 42.9%,

32.7%, 31.3%, 46.9%. Therefore, over the whole sample, the persistence in the choice of

risk levels is particularly high for fund investing to a large extent in small caps (high SMB

factor). However, this finding is consistent with prior expectations because the choice of

risk level and its persistence is, at least partially, induced by a fund’s policy.

For the tracking error against the S&P 500, the diagonal transition probabilities are 61.9%,

36.8%, 32.5%, 62.4%. Therefore, the persistence of active and passive portfolio

management measured as the deviation from the index is substantial.

Overall, we find strong evidence for persistence in the choice of risk levels. In particular,

funds with a very high and very low exposure to specific risk factors show a high degree of

persistence. However, these results are not surprising because a number of factors and, in

particular, institutional restrictions induce a persistent behavior.

19

3.3. The Impact of Prior Performance

In this section, we turn to the impact of prior performance on the risk taking of mutual fund

managers. The complete empirical results are included in the appendix in Tables A1 and

A2. Due to the large amount of empirical data, we focus the interpretation on the results

shown in the Tables 4 and 5.

Table 4 shows the difference in transition probabilities for top- and poor-performing

mutual funds for risk-adjusted returns using the S&P 500 as benchmark index. For the

volatility, we find that fund managers with a poor performance, in general, strongly

decrease their portfolio volatility in the following calendar year. For example, the

difference in transition probabilities for funds with a volatility in the highest quartile is

18.9%. Only successful funds with a low volatility (in Q1) increase the volatility in the

following calendar year.

For the beta, the evidence is mixed. Funds with an exposure to the market risk above the

median (Q3 and Q4) take more market risk in the following calendar year, e.g. the

difference in transition probabilities for Q4 to Q4 is 9.8%. This indicates that successful

fund managers are 9.8% more likely to maintain their risk level than for unsuccessful

managers. In contrast, unsuccessful managers with a beta in Q1 increase their market risk

exposure significantly. For example, the difference in transition probabilities from Q1 to

Q2 is -8.4%. This indicates than poor-performing funds are 8.4% more likely to increase

their market risk exposure.

The style factors are interpreted as follows. A high loading (e.g. in Q4) on the value factor

(HML) indicates that a fund invests in value stocks. Vice versa, a low factor loading on the

value factor (Q1) is interpreted as an investment in growth stocks. A fund investing in

20

small caps shows a high exposure to the size factor (SMB) and a large-cap funds shows an

exposure to in the first quartile. Funds investing in momentum stocks have a high loading

on the UMD factor (Q4) and contrarian funds a low loading on UMD.

For the HML, SMB and UMD factor, there is overwhelming evidence for increased risk

taking by successful fund managers. Successful fund managers invest in the future heavily

in value stocks, small caps and momentum stocks.

For the loading on the value premium (HML), all differences in transition probabilities

ending in Q1 are negative and ending in Q4 are positive. For example, successful funds

having invested in growth stocks are 6.7% more likely to switch to a substantial value

investment (transition element Q1 to Q4) and poor-performing funds are 8.5% more likely

to continue the unsuccessful growth stock investments (transition element Q1 to Q1).

For the size exposure, the change in behavior is even stronger. All successful manager

regardless of their prior size exposure invest substantially in small caps. For example,

successful managers who have previously invested primarily in large caps are 6.8% more

likely to invest substantially in small caps than their unsuccessful counterparts (transition

probability Q1 to Q4).

For the momentum exposure, the previously observed pattern, i.e., increased risk-taking by

successful mangers, is also observable, but less evident. Material changes are observable

for fund managers having previously neglected momentum stocks (Q1). If these fund

managers achieved a good-performance they are 9.5% more likely to invest in the future

heavily in momentum stocks than poor-performing managers (transition probability Q1 to

Q4).

The tracking error as a measure of active portfolio management validates the findings for

other variables. Successful managers increase, in general, the tracking error and poor-

21

performing fund managers tend to decrease it. For example, successful fund managers are

17.0% more likely to maintain a tracking error in the highest quartile (Q4) compared to

unsuccessful fund managers.

Next, we analyze how portfolio managers respond to prior performance measured with the

Carhart four-factor model. As shown by Carhart (1997), the four-factor model explains a

high proportion of the cross sectional variance of the performance of mutual funds. After

accounting for a fund’s exposure against value and growth stocks, large and small caps,

and momentum stocks, he finds little evidence for persistence in the performance of mutual

funds.

Overall, a comparison of the results for returns risk-adjusted with the Carhart four-factor

model on the one side and with a one factor model, i.e. the S&P 500 as benchmark indicate

differences as shown in Table 5. While Table 4 indicates substantial and significant

changes in behaveor, in particular for the first and fourth quartiles, the results for the

Carhart four-factor model are less evident.

For the volatility, we find, in general, no significant evidence for a change in behavior.

However, there is little evidence for increased risk-taking by low performing funds. For the

Beta, the data confirm our previous findings. A good performance induces increased taking

of market risk in the following period. However, most findings are statistically

insignificant.

For the HML, SMB and UMD factor, the results differ somehow from previous findings

and are mixed. For HML, we find clear statistically significant pattern. For SMB findings

reverse. For UMD, results are ambiguous. For the Carhart four-factor, we find strong

evidence that successful fund managers abstain from small caps. For example, the

22

transition probability from Q4 to Q4 is -3.4%. This indicates that successful fund managers

tend to reduce their exposure to small caps compared to poor-performing fund managers.

After controlling for a fund’s style, we find that unsuccessful funds tend to change their

momentum strategy into different directions. For example, the transition probability Q2 to

Q2 is significantly positive, having a value of 8.1%. This indicates that successful funds

are 8.1% more likely to choose a comparable momentum level. Unsuccessful funds change

their strategy into both directions, a contrarian strategy and a stronger momentum strategy.

For the tracking error, superior performance induces a more passive investments style after

controlling for a fund’s style. However, this findings is only significant for the first quartile

transition probabilities.

3.4. Discussion

In this section, we discuss the results of our econometric analysis with respect to

theoretical models and with respect to other empirical findings.

Existing literature trying to explain the behavior of mutual fund managers (e.g., Brown et

al. (1996), Chevalier & Ellison (1997), Carpenter (2000), Busse (2001), and Carhart,

Kaniel, Musto & Reed (2002)) has focussed on incentives faced by mutual fund managers.

Incentives in the mutual fund industry are primarily driven by two factors, by the

compensation schemes and the behavior of investors. Standard compensation schemes in

the mutual fund industry are convex, i.e., fund managers take part in the positive

performance of their funds by receiving bonuses while they usually do not take part in the

negative performance of their funds. Portfolio managers have a call option on the portfolio

they are managing. Moreover, as a second important incentive factor, the intertemporal

behavior of investors increases the effect of convex compensation schemes. Investors

23

allocate a large proportion of new capital to funds with a high performance in the previous

period whereas they do not withdraw invested capital from poorly performing funds.

Therefore, if a manager’s salary depends on the assets under management, investors’

behavior induces a convex relation between fund performance and fund size. Overall, both

patterns lead in theory to excessive risk taking by mutual fund managers.

However, in our empirical analysis, we were unable to find evidence for such a behavior.

In general, these findings can be explained by a different setting in this study compared to

other empirical studies. In particular, we use a large sample of funds over a longer time

period and different measures of risk and return. Moreover, we impose less restrictive

assumptions for the empirical analysis, e.g., we do not assume any linear relation or a

normal distribution. In particular, Brown et al. (1996) solely focus on volatility as a

measure of risk and only use a rather small sample of funds, funds focussing on growth

stocks, over a time period of 15 years. Their analysis focusses on mid-year effects and they

find that funds performing poorly by mid-year increase their volatility for the rest of the

year. Busse (2001) uses a very similar methodology and the same data set as in Brown et

al. (1996), but with a daily frequency, and finds that the fund’s intra-year change is

attributable to changes in the volatility of common stocks and not related to changing

factor exposures or residual risk. Similar, Chevalier & Ellison (1997) analyze the impact of

past performance on fund flows using a semi-parametric approach. Their results confirm

prior expectations, i.e., the flow-performance relation creates incentives for fund managers

to adjust the riskiness of the fund depending on mid-year performance.

How can we explain increased risk taking after years of good performance and decreased

risk taking after years of poor performance? Our explanation is two-sided. First, poor-

performing managers follow a more passive strategy to minimize the risk of their own

24

future replacement. Relative performance and not absolute performance is relevant.

Second, successful managers take more risk, because they have become more confident in

their own skills. Success creates confidence. Basically, our analysis shows that a

combination of the models by Lynch & Musto (2003) for unsuccessful managers and Berk

& Green (2004) for successful managers describe the data best.

Lynch & Musto (2003) propose a model in which strategy changes only occur after periods

of bad performance. However, a priori, their model does not say how the strategy changes.

In their empirical analysis, they find evidence for a change of factor loadings. They find

that poor performers seem to increase their UMD loading and decrease their HML loading.

Neither market beta nor SMB loading is systematically affected by fund performance. The

different results by Lynch & Musto (2003) in comparison to our analysis might be due to

the shorter sample period in the analysis by Lynch & Musto (2003) and to the use of a non-

linear model in our analysis. Our analysis shows that, after a period of bad performance,

managers choose a passive investments style (lower tracking error), take less market risk,

decrease their exposure to value, and increase their exposure to large caps, and stocks with

a low momentum effect.

Berk & Green (2004) propose a model that incorporates two important features. First,

performance is not persistent, i.e., active portfolio managers do not outperform passive

benchmarks on average and, second, funds flows rationally respond to past performance.

In particular, they assume that investors behave as Bayesians, i.e., they update their beliefs

about a fund manager’s skill based on observed returns and prior beliefs. The same

argumentation can be applied to a manager’s belief about his own skills. Starting with

some prior confidence on his own skills, he learns about his skills and uses his skills in the

following time period. If skills require specific trading strategies, e.g., a fund manager has

25

special skills in selecting small caps or value stocks, an empirical analysis will show

patterns as we have observed.

Mutual fund managers behavior is more complex than assumed by theoretical models,

which usually capture only one aspect of the actual behavior. According to our analysis, a

combination of the models by Lynch & Musto (2003) and Berk & Green (2004) describes

the findings best.

4. Conclusion

How do mutual fund managers react to past performance? Theory suggests that good-

performing mutual fund managers reduce their risk level and poor-performing managers

take more risk because they do not bear the downside risk. However, such a behavior

might be unrealistic in actual life due to restrictions fund managers are facing, such as

tracking error restrictions. Moreover, other factors affecting the behavior of mutual fund

managers might be more important than pure compensation maximization. For example, if

relative performance is more important than absolute performance, they tend to take only

little idiosyncratic risk compared to their relevant benchmark.

In this paper, we analyze a large sample of US investments funds for a period of 20 years.

For each year, we compute different measures of style and risk. Overall, our analysis

extends the existing literature in a number of ways. In contrast to existing studies, we do

not solely focus on volatility as a measure of risk. We take also other measures such as the

beta and the tracking error, and style measures such as the high-minus-low (HML) factor,

the small-minus-big (SMB) factor, and the momentum (UMD) factor into account.

Furthermore, we use a robust, non-parametric approach and are therefore able to capture a

26

wide range of non-linear and asymmetric patterns because we do not impose any restrictive

distributional assumptions. Concerning the data basis, we do not use only a sub-group of

mutual funds, but a complete set of all US equity funds with a long time period of data.

Our analysis does not lend any support for the hypothesis that poor-performing fund

managers increase the risk level. We find that prior performance has a positive impact on

the choice of the risk level, i.e., successful fund managers take more risk in the following

time period. In particular, they increase the volatility, the beta and the tracking error and

assign a higher proportion of their portfolio to value stocks, small firms and momentum

stocks. Overall, poor performing fund managers switch to passive strategies. Unsuccessful

managers decrease the level of idiosyncratic risk and follow the relevant benchmark more

closely.

References

Berk, J. & Green, R. (2004), ‘Mutual fund flows and performance in rational markets’, Journal of Political Economy 112, 1269–1295. Borgelt, C. & Kruse, R. (2002), Graphical Models, John Wiley & Sons, New York. Brown, K., Harlow, W. & Starks, L. (1996), ‘Of tournaments and temptations: An analysis of managerial incentives in the mutual fund industry’, Journal of Finance 51, 85–110. Busse, J. (2001), ‘Another look at mutual fund tournaments’, Journal of Financial and Quantitative Analysis 36, 53–73. Carhart, M. (1997), ‘On persistence in mutual fund performance’, Journal of Finance 52, 57–82. Carhart, M. M., Kaniel, R., Musto, D. K. & Reed, A. V. (2002), ‘Learning for the tape: Evidence of gaming behavior in equity mutual funds’, Journal of Finance 68, 661–693. Carpenter, J. (2000), ‘Does option compensation increase managerial risk appetite?’, Journal of Finance 55, 2311–2331.

27

Chevalier, J. & Ellison, G. (1997), ‘Risk taking by mutual funds as a response to incentives’, Journal of Political Economy 105, 1167–1200. Cowell, R. G., Dawid, A. P., Lauritzen, S. L. & Spiegelhalter, D. J. (2003), Probabilistic Networks and Expert Systems, Springer, New York. Dangl, T., Wu, Y. & Zechner, J. (2004), ‘Mutual fund flows and optimal manager replacement’, Working Paper. University of Vienna. Daniel, K., Grinblatt, M., Titman, S. & Wermers, R. (1997), ‘Measuing mutual fund performance with characteristic-based benchmarks’, Journal of Finance 52, 1035–1058. Deli, D. N. (2002), ‘Mutual fund advisory contracts: An empirical investigation’, Journal of Finance 57, 109–133. Duda, R. O., Hart, P. E. & Stork, D. G. (2001), Pattern Classification, John Wiley & Sons, New York. Fama, E. & French, K. (1993), ‘Common risk factors in the returns on stocks and bonds’, Journal of Financial Economics 33, 3–57. Ghahramani, Z. (2001), ‘An introduction to Hidden Markov Models and Bayesian networks’, International Journal of Pattern Recognition and Artificial Intelligence 15, 9–42. Jensen, F. J. (2001), Bayesian Networks and Decision Graphs, Springer, New York. Jordan, M. I., Ghahramani, Z. & Saul, L. K. (1997), ‘An introduction to variational methods for graphical models’, Working Paper . Korb, K. B. & Nicholson, A. E. (2004), Bayesian Artifical Intelligence, Chapmann & Hall / CRC, New York. Kothari, S. & Warner, J. B. (2001), ‘Evaluating mutual fund performance’, Journal of Finance 56, 1985–2010. Lynch, A. & Musto, D. (2003), ‘How investors interpret past fund returns’, Journal of Finance 58, 2033–2058. Neapolitan, R. E. (2004), Learning Bayesian Networks, Prentice Hall, Upper Saddle River. Pearl, J. (1982), Reverend Bayes on inference engine: A distributed hierarchical approach, in ‘Proceedings, Cognitive Science Society’, Ablex, Greenwich, CT, pp. 329–334. Pearl, J. (2000), Causality: Models, Reasoning and Inference, Cambridge University Press, Cambridge.

28

Schmidt, B. (2003), ‘Hierarchical Bayesian models, holding propensity and the mutual fund flow-performance relation’, Working Paper. University of Chicago. Wermers, R. (2000), ‘Mututal fund performance: An empirical decomposition into stock-picking, style, transactions costs, and expenses’, Journal of Finance 55, 1655–1695.

29

Exhibits

Figure 1

Illustration of a Dynamic Bayesian Network The three main issues in the area of Bayesian Networks are (1) the question of representation, i.e., what is a Bayesian Network, (2) probabilistic inference, i.e., the updating of probability distributions for a query node given evidence for a particular node, and (3) the estimation of parameters for conditional probability distributions for a Bayesian Network based on sample data. Representation: A Bayesian Network consists of a set of variables, usually denoted as nodes (illustrated as circles). The arcs represent conditional dependencies in the network between nodes. Inference: Probabilistic inference denotes the updating of probability distributions for a query node given some evidence (posterior distribution). The Graph illustrates the probability updating for the query node TRT, i.e., the tracking error in Period T + 1, conditional on the evidence that RT, i.e., return in period T, was high. Learning: To learn the parameters for conditional probability distributions in a Bayesian Network maximum likelihood methods are applicable.

30

Figure 2 Dynamic Bayesian Network used in the empirical analysis

For a number of different measures of risk, we analyze the joint effect of the risk level in T and the risk-adjusted return in T on the choice of the risk level in T+1. The data set was provided by Reuters Lipper and consists of 1923 funds with return data from 1984 to 2004. As measures of risk we used: volatility (STD), the beta with respect to the market portfolio (MRP), the factor loading on the value premium (HML), the factor loading on the size premium (SMB), the factor loading on the momentum premium (UMD), and the tracking error (TE). A return is denoted by R. The tracking error and the beta have been computed with respect to the S&P 500 and the CRSP market portfolio using a one factor and a four-factor model, respectively (denoted as TESP500, TECarhart, MRPSP500, MRPCarhart. Similar, return R has been computed on a raw basis (without risk-adjustment), risk-adjusted in a one factor model with respect to the S&P 500 and to the Carhart (1997) four factor model (denoted as RRaw, RSP500, and RCarhart).

31

Table 1 Descriptive statistics for annual continuously compounded returns

The table shows the descriptive statistics for annual continuously compounded returns for all funds existing in one particular year. The data set is from Reuters Lipper.

Year Funds Mean Std. Skew Kurt Min 25% 50% 75% Max

1985 191 18.87 7.37 -0.01 3.67 -3.23 14.18 19.43 23.23 43.561986 216 0.49 10.80 -0.20 5.28 -48.43 -5.64 0.55 8.00 45.381987 243 -13.49 12.46 -0.23 5.87 -60.19 -19.86 -13.13 -5.38 42.811988 284 9.46 8.12 0.08 4.52 -15.31 4.27 9.58 14.36 46.101989 305 15.31 8.38 0.01 3.44 -9.40 10.36 14.95 20.95 41.451990 326 -11.70 8.79 -0.63 3.76 -41.48 -16.63 -10.80 -5.52 11.511991 350 26.28 12.08 -1.29 16.97 -75.62 18.99 25.39 32.88 63.081992 387 3.52 8.02 -0.72 7.93 -48.47 -0.39 3.59 7.29 30.101993 452 5.34 8.53 -1.14 10.06 -51.79 0.49 5.73 10.44 36.901994 539 -6.52 7.30 -0.54 5.32 -41.25 -10.34 -5.95 -2.41 21.081995 622 20.44 8.72 -0.36 5.07 -22.82 15.67 21.01 25.75 48.421996 701 9.39 9.09 -1.07 9.59 -61.24 4.93 9.56 14.72 44.061997 822 11.36 11.24 -1.18 7.27 -56.69 5.48 12.89 18.44 49.911998 977 6.57 15.34 -0.12 3.58 -48.04 -3.24 6.84 16.87 58.221999 1114 17.05 22.76 1.07 5.02 -53.06 1.48 13.49 27.79 136.392000 1233 -12.14 22.51 -1.52 9.89 -188.63 -23.45 -10.09 2.68 40.862001 1369 -11.83 16.82 -0.30 4.28 -90.89 -21.35 -12.61 -1.30 45.782002 1478 -26.35 11.34 -0.66 5.64 -103.74 -32.59 -26.08 -19.18 14.082003 1478 28.26 8.98 0.42 12.28 -58.39 22.29 26.70 33.06 97.56

32

Table 2 Transition probabilities between risk and return

The table shows the transition probabilities between different measures of risk and the subsequent return. * denotes a value statistically different from 0.25 on a 95% level and ** on a 99% level. Standard errors have been computed by bootstrapping. The table is interpreted as follows. Suppose a fund has in one particular year a volatility in the lowest quartile among all other funds. Then, the fund has a probability of 11.8% (first row, firstcolumn) to achieve a risk-adjusted return in the lowest quartile among all other funds. Analogously, the chance is 30.8% (first row, fourth column) to obtain a risk-adjusted return in the highest quartile among all other funds.

tofrom Q1 Q2 Q3 Q4STDT Q1 11.8%* 28.9%** 28.3%** 30.8%**

Q2 15.5%* 25.1% 30.6%** 28.6%**Q3 17.9%* 21.7%* 25.0% 34.4%**Q4 22.8%* 16.6%* 19.0%* 41.4%**

MRPSP500,T Q1 16.0%* 18.6%* 22.6%* 42.6%**Q2 15.1%* 22.4%* 27.8%* 34.5%**Q3 16.5%* 25.7% 27.4%** 30.2%**Q4 21.1%* 21.8%* 23.9% 33.0%**

HMLT Q1 24.0% 23.0%* 19.4%* 32.6%**Q2 15.7%* 28.0%** 30.0%** 26.2%Q3 13.1%* 24.7% 30.3%** 31.8%**Q4 17.8%* 15.6%* 22.2%* 44.2%**

SMBT Q1 21.7%* 28.2%** 25.7% 24.2%Q2 14.5%* 30.4%** 29.9%** 25.0%Q3 13.3%* 24.2% 29.6%** 32.7%**Q4 19.5%* 14.8%* 21.1%* 44.4%**

UMDT Q1 33.7%** 30.7%** 18.4%* 17.0%*Q2 20.3%* 32.1%** 26.5%** 21.0%*Q3 12.9%* 22.0%* 32.0%** 33.0%**Q4 12.9%* 14.3%* 23.2%* 49.4%**

TESP500,T Q1 14.3%* 29.2%** 30.8%** 25.5%Q2 18.1%* 25.1% 27.7%** 29.0%**Q3 18.0%* 19.0%* 22.5%* 40.3%**Q4 20.7%* 13.9%* 19.4%* 45.8%**

RSP500,T

33

Table 3 Transition probabilities between risk in T and T+1

The table shows the transition probabilities between different measures of risk in T and T+1. XXX denotes that the target (column) variable is the same variable as in the appropriate row. The table shows the results for the model with risk-adjustment using the S&P 500. * denotes a value statistically different from 0.25 on a 95% level and ** on a 99% level. Standard errors have been computed by bootstrapping. The table is interpreted as follows. Suppose a fund has in one particular year a volatility in the lowest quartile among all other funds. Then, the fund manger has selected with a probability of 50.4% (first row, first column) a return volatility in the lowest quartile among all funds in the next calendar year. Analogously, the chance is 2.6% (first row, fourth column) to select a volatility on the highest quartile among all other funds in the next calendar year.

to

from Q1 Q2 Q3 Q4STDT Q1 50.4%** 32.9%** 14.0%* 2.6%*

Q2 24.4% 37.8%** 29.7%** 7.9%*Q3 8.4%* 22.8% 41.2%** 27.4%Q4 1.2%* 6.3%* 22.2%* 70.1%**

MRPSP500,T Q1 46.0%** 28.7%** 17.6%* 7.5%*Q2 25.4% 31.0%** 27.9%** 15.5%*Q3 11.8%* 25.5% 36.4%** 26.1%Q4 4.7%* 13.8%* 27.0% 54.1%**

HMLT Q1 49.7%** 24.6% 15.5%* 9.9%*Q2 26.4% 34.3%** 25.0% 13.3%*Q3 14.9%* 27.7%** 32.4%** 24.8%Q4 11.5%* 15.4%* 28.5%** 44.4%**

SMBT Q1 49.3%** 29.5%** 14.7%* 6.4%*Q2 37.8%** 33.1%** 18.9%* 10.0%*Q3 15.4%* 17.7%* 35.1%** 31.7%**Q4 4.2%* 5.8%* 21.1%* 68.7%**

UMDT Q1 42.9%** 25.2% 18.0%* 13.7%*Q2 26.0% 32.7%** 24.5% 16.1%*Q3 16.1%* 27.8%** 31.3%** 24.6%Q4 11.4%* 15.2%* 26.3% 46.9%**

TESP500,T Q1 61.9%** 25.5% 9.1%* 3.4%*Q2 22.5% 36.8%** 25.4% 15.2%*Q3 7.4%* 23.9% 32.5%** 36.0%**Q4 1.7%* 9.8%* 26.0% 62.4%**

XXXT

M: RSP500,T

34

Table 4 Impact of prior performance on the choice of risk level (returns

adjusted with the S&P 500) The table shows the difference in transition probabilities between different measures of risk in T and T+1 for funds with a performance in the highest quarter in T and for funds with a performance in the lowest quarter in T. XXX denotes that the target (column) variable is the same variable as in the appropriate row. The table shows the results for returns adjusted with the S&P 500. * denotes a value statistically different from 0 on a 95% level and ** on a 99% level. Standard errors have been computed by bootstrapping. The results displayed in this table base on a risk-adjusted return using a one factor model and the S&P 500 as market portfolio. The results in the table have been computed as follows. Top-performing funds are funds with a return in the fourth quartile in one calendar year, poor-performing funds are funds with a return in the first quartile in one calendar year. Both groups have different transition matrices for the risk level in the next year. In this table, we show the difference of element-by-element subtraction of the transition matrices. The transition matrix of poor-performing funds has been subtracted from the matrix of top-performing funds. Therefore, positive elements indicate that the transition probability for top-performing funds was higher than for poor-performing funds and vice-versa.

to

from Q1 Q2 Q3 Q4STDT Q1 6.3% -1.5% -1.1% -3.6%**

Q2 -10.3%** -0.9% 7.4%* 3.8%*Q3 -8.5%** -9.0%** 0.0% 17.0%**Q4 -2.1%** -6.0%** -10.7%** 18.9%**

MRPSP500,T Q1 13.3%** -8.4%** -1.8% -3.0%**Q2 3.3% -6.8%** 3.3% 0.0%Q3 -5.8%** -4.7%** 1.7% 8.8%**Q4 -5.0%** -1.8% -3.0% 9.8%**

HMLT Q1 -8.5%** -3.6% 5.4%** 6.7%**Q2 -8.4%** -0.5% 2.3% 6.6%**Q3 -7.1%** -4.1% 5.8%* 5.4%*Q4 -9.8%** -0.5% 3.2%* 7.1%**

SMBT Q1 -6.7%* -3.1% 3.0% 6.8%**Q2 -11.4%** -1.8% 5.5%* 7.7%**Q3 -13.1%** -7.9%** 5.0%* 16.0%**Q4 -6.5%** -5.8%** -1.2% 13.6%**

UMDT Q1 -13.0%** -5.2%* 8.7%** 9.5%**Q2 -3.4% -1.4% 2.0% 2.7%Q3 -8.7%** 1.5% 9.5%** -2.2%Q4 -6.6%** -1.9% 5.3%** 3.2%

TESP500,T Q1 13.0%** -11.5%** -4.1% 2.7%*Q2 -5.8%* -4.3% 1.7% 8.4%**Q3 -6.6%** -5.0%* 0.9% 10.6%**Q4 -3.1%** -9.3%** -4.5%* 17.0%**

XXXT+1

Difference

35

Table 5 Impact of prior performance on the choice of risk level (returns

adjusted with the Carhart model) The table shows the difference in transition probabilities between different measures of risk in T and T+1 for funds with a performance in the highest quarter in T and for funds with a performance in the lowest quarter in T. XXX denotes that the target (column) variable is the same variable as in the appropriate row. The table shows the results for returns adjusted with the Carhart model. * denotes a value statistically different from 0 on a 95% level and ** on a 99% level. Standard errors have been computed by bootstrapping. The results in the table have been computed as follows. Top-performing funds are funds with a return in the fourth quartile in one calendar year, poor-performing funds are funds with a return in the first quartile in one calendar year. Both groups have different transition matrices for the risk level in the next year. In this table, we show the difference of element-by-element subtraction of the transition matrices. The transition matrix of poor-performing funds has been subtracted from the matrix of top-performing funds. Therefore, positive elements indicate that the transition probability for top-performing funds was higher than for poor-performing funds and vice-versa.

to

from Q1 Q2 Q3 Q4STDT Q1 7.0% 1.6% -5.8% -2.8%**

Q2 2.1% -3.8% 2.6% -1.0%Q3 1.0% 1.7% -0.1% -2.6%Q4 0.0% 0.0% -1.2% 1.4%

MRPSP500,T Q1 -0.2% -2.1% 4.0%* -1.6%Q2 -6.8%** -3.7% 7.6%** 2.8%Q3 -3.9%* -7.5%** 2.7% 8.6%*Q4 -2.3%** -1.2% 0.0% 2.9%

HMLT Q1 -3.4% -1.5% 2.4%* 2.5%*Q2 -5.9%** 7.1%** 2.1% -3.3%Q3 -4.6%* 2.4% 0.4% 1.7%Q4 0.0% -0.3% 0.0% -0.1%

SMBT Q1 15.5%** 1.4% -10.3%** -6.6%**Q2 7.4%** -0.3% -3.2% -3.7%*Q3 -3.9%* 0.0% -0.1% 4.1%Q4 -0.5% -0.2% 4.1%* -3.4%*

UMDT Q1 1.7% 1.2% 1.9% -4.9%**Q2 -2.0% 8.1%** -0.4% -5.5%**Q3 -7.2%** 0.6% 11.2%** -4.0%Q4 -2.6%* -0.5% 5.9%** -2.6%

TESP500,T Q1 30.4%** -6.9%** -18.9%** -4.4%**Q2 3.6% 3.6% -3.2% -4.0%Q3 -1.8%* 5.9%* -2.6% -1.4%Q4 -0.1% 0.0% 1.6% -1.5%

XXXT+1

Difference

36

Table A1

Impact of prior performance on the choice of risk level (returns adjusted with the S&P 500) The table shows the transition probabilities between different measures of risk in T and T+1 for funds with a performance in the highest quarter in T (first four columns), for funds with a performance in the lowest quarter in T (middle four columns), and the difference between these transition probabilities. XXX denotes that the target (column) variable is the same variable as in the appropriate row. The table shows the results for returns adjusted with the S&P 500. For the left and middle set of columns, * denotes a value statistically different from 0.25 on a 95% level and ** on a 99% level. For the difference between transition probabilities in the right set of columns, the null hypothesis is 0.00, i.e., we test whether this difference is statistically different from 0. Standard errors have been computed by bootstrapping.

to

from Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4STDT Q1 50.6%** 31.1%** 15.5%* 2.7%* 44.2%** 32.7%* 16.6%* 6.3%* 6.3% -1.5% -1.1% -3.6%

Q2 22.3% 35.4%** 29.8%* 12.3%* 32.7%* 36.3%** 22.4% 8.4%* -10.3%** -0.9% 7.4%* 3.8%Q3 4.8%* 16.1%* 39.5%** 39.4%** 13.4%* 25.2% 38.9%** 22.3% -8.5%** -9.0%** 0.0% 17.0Q4 0.3%* 2.6%* 14.3%* 82.5%** 2.5%* 8.6%* 25.1% 63.6%** -2.1%** -6.0%** -10.7%** 18.9

MRPSP500,T Q1 54.3%** 23.8% 14.8%* 6.9%* 40.9%** 32.2%** 16.7%* 10.0%* 13.3%** -8.4%** -1.8% -3.0%Q2 31.0%** 26.5% 26.0% 16.3%* 27.6% 33.3%** 22.6% 16.3%* 3.3% -6.8%** 3.3% 0.0Q3 13.8%* 23.3% 30.6%** 32.1%** 19.6%* 28.1% 28.8% 23.3% -5.8%** -4.7%** 1.7% 8.8%Q4 3.9%* 11.3%* 22.1%* 62.5%** 8.9%* 13.2%* 25.1% 52.6%** -5.0%** -1.8% -3.0% 9.8%

HMLT Q1 46.1%** 21.6%* 17.8%* 14.3%* 54.7%** 25.2% 12.4%* 7.5%* -8.5%** -3.6% 5.4%** 6.7%Q2 26.0% 28.8% 24.5% 20.4%* 34.5%** 29.4%** 22.1%* 13.8%* -8.4%** -0.5% 2.3% 6.6%Q3 14.3%* 18.7%* 31.9%** 34.9%** 21.4% 22.8% 26.1% 29.5%** -7.1%** -4.1% 5.8%* 5.4%Q4 7.7%* 13.3%* 26.2% 52.5%** 17.6%* 13.9%* 2.0% 45.4%** -9.8%** -0.5% 3.2%* 7.1%

SMBT Q1 38.7%** 24.7% 22.2%* 14.2%* 45.5%** 27.9% 19.2%* 7.3%* -6.7%* -3.1% 3.0% 6.8%Q2 24.0% 29.2%** 25.3% 21.3%* 35.5%** 31.1%** 19.7%* 13.6%* -11.4%** -1.8% 5.5%* 7.7%Q3 7.2%* 10.7%* 36.0%** 45.9%** 20.3%* 18.7%* 30.9%** 29.9%** -13.1%** -7.9%** 5.0%* 16.0Q4 0.8%* 2.3%* 19.1%* 77.6%** 7.4%* 8.1%* 20.4%* 63.9%** -6.5%** -5.8%** -1.2% 13.6

UMDT Q1 33.2%** 17.6%* 24.9% 24.1% 46.3%** 22.8%* 16.2%* 14.6%* -13.0%** -5.2%* 8.7%** 9.5%Q2 25.3% 24.6% 25.3% 24.5% 28.8%* 26.1% 23.1% 21.8%* -3.4% -1.4% 2.0% 2.7Q3 13.3%* 23.8% 32.2%** 30.6%** 22.1% 22.2% 22.6% 32.8%** -8.7%** 1.5% 9.5%** -2.2Q4 8.3%* 11.4%* 25.1% 55.0%** 14.9%* 13.3%* 19.8%* 51.8%** -6.6%** -1.9% 5.3%** 3.2

TESP500,T Q1 55.0%** 27.1% 11.4%* 6.3%* 42.0%** 38.7%** 15.5%* 3.6%* 13.0%** -11.5%** -4.1% 2.7%Q2 13.1%* 35.0%** 27.3% 24.3% 19.0%* 39.4%** 25.0% 15.8%* -5.8%* -4.3% 1.7% 8.4%Q3 2.1%* 19.7%* 33.4%** 44.5%** 8.8%* 24.8% 32.5%** 33.8%** -6.6%** -5.0%* 0.9% 10.6Q4 0.3%* 4.7%* 20.8%* 74.0%** 3.5%* 14.0%* 25.4% 56.9%** -3.1%** -9.3%** -4.5%* 17.0

XXXT+1

DifferenceXXXT+1

RSP500,T=4Q RSP500,T=1QXXXT+1

Table A2 Impact of prior performance on the choice of risk level (returns adjusted with the Carhart model)

The table shows the transition probabilities between different measures of risk in T and T+1 for funds with a performance in the highest quarter in T (first four columns), for funds with a performance in the lowest quarter in T (middle four columns), and the difference between these transition probabilities. XXX denotes that the target (column) variable is the same variable as in the appropriate row. The table shows the results for returns adjusted with the Carhart model. For the left and middle set of columns, * denotes a value statistically different from 0.25 on a 95% level and ** on a 99% level. For the difference between transition probabilities in the right set of columns, the null hypothesis is 0.00, i.e., we test whether this difference is statistically different from 0. Standard errors have been computed by bootstrapping.

to

from Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4STDT Q1 47.3%** 32.2%* 17.5%* 2.9%* 40.2%** 30.5%* 23.4% 5.8%* 7.0% 1.6% -5.8% -2.8%*

Q2 23.6% 32.5%** 33.4%** 10.4%* 21.4% 36.3%** 30.7%* 11.4%* 2.1% -3.8% 2.6% -1.0%Q3 8.1%* 22.2% 38.4%** 31.1%** 7.0%* 20.5%* 38.5%** 33.8%** 1.0% 1.7% -0.1% -2.6%Q4 1.3%* 6.4%* 20.7%* 71.5%** 1.5%* 6.4%* 21.9%* 70.0%** 0.0% 0.0% -1.2% 1.4%

MRPSP500,T Q1 42.3%** 26.8% 21.1%* 9.5%* 42.6%** 29.0%* 17.0%* 11.2%* -0.2% -2.1% 4.0%* -1.6%Q2 19.0%* 28.2% 31.3%** 21.3% 25.8% 31.9%** 23.6% 18.5%* -6.8%** -3.7% 7.6%** 2.8%Q3 12.0%* 20.5%* 32.9%** 34.5%** 15.9%* 28.0% 30.1%** 25.8% -3.9%* -7.5%** 2.7% 8.6%*Q4 4.8%* 12.3%* 23.4% 59.3%** 7.1%* 13.5%* 22.8% 56.4%** -2.3%** -1.2% 0.0% 2.9%

HMLT Q1 50.9%** 24.1% 14.4%* 10.4%* 54.3%** 25.6% 12.0%* 7.9%* -3.4% -1.5% 2.4%* 2.5%*Q2 29.1%* 32.9%** 24.6% 13.2%* 35.0%** 25.8% 22.4%* 16.5%* -5.9%** 7.1%** 2.1% -3.3%Q3 16.4%* 25.1% 31.4%** 26.9% 21.0% 22.7% 30.9%** 25.1% -4.6%* 2.4% 0.4% 1.7%Q4 16.2%* 14.8%* 26.0% 42.8%** 16.1%* 15.1%* 25.5% 43.0%** 0.0% -0.3% 0.0% -0.1%

SMBT Q1 55.9%** 26.4% 11.1%* 6.4%* 40.3%** 24.9% 21.5%* 13.1%* 15.5%** 1.4% -10.3%** -6.6%*Q2 37.3%** 29.9%** 19.3%* 13.3%* 29.9%** 30.3%** 22.5% 17.1%* 7.4%** -0.3% -3.2% -3.7%Q3 10.6%* 14.8%* 34.2%** 40.2%** 14.6%* 14.8%* 34.4%** 36.0%** -3.9%* 0.0% -0.1% 4.1%Q4 4.1%* 6.0%* 23.4% 66.3%** 4.6%* 6.2%* 19.2%* 69.8%** -0.5% -0.2% 4.1%* -3.4%

UMDT Q1 46.4%** 22.4% 17.8%* 13.2%* 44.6%** 21.2%* 15.9%* 18.1%* 1.7% 1.2% 1.9% -4.9%*Q2 26.5% 35.0%** 22.4% 15.9%* 28.7% 26.9% 22.8% 21.5%* -2.0% 8.1%** -0.4% -5.5%*Q3 16.1%* 25.3% 32.9%** 25.5% 23.3% 24.7% 21.6% 30.2%* -7.2%** 0.6% 11.2%** -4.0%Q4 12.7%* 14.3%* 27.8%* 45.0%** 15.3%* 14.9%* 21.9%* 47.7%** -2.6%* -0.5% 5.9%** -2.6%

TESP500,T Q1 67.1%** 24.0% 5.3%* 3.4%* 36.7%** 31.0%* 24.3% 7.9%* 30.4%** -6.9%** -18.9%** -4.4%*Q2 18.7%* 37.8%** 25.5% 17.8%* 15.0%* 34.2%** 28.8% 21.9% 3.6% 3.6% -3.2% -4.0%Q3 4.2%* 24.9% 31.9%** 38.7%** 6.1%* 19.0%* 34.6%** 40.2%** -1.8%* 5.9%* -2.6% -1.4%Q4 1.6%* 10.0%* 26.7% 61.5%** 1.8%* 9.9%* 25.1% 63.0%** -0.1% 0.0% 1.6% -1.5%

XXXT+1

DifferenceXXXT+1

RCarhart,T=4QXXXT+1

RCarhart,T=1Q